Problem: $\dfrac{d}{dx}[-4x^3-\sin(x)]=$
Explanation: The expression to differentiate includes $\sin(x)$. Remember that the derivative of $\sin(x)$ is $\cos(x)$. Put another way, $\dfrac{d}{dx}[\sin(x)]=\cos(x)$. $\begin{aligned} &\phantom{=}\dfrac{d}{dx}[-4x^3-\sin(x)] \\\\ &=-4\dfrac{d}{dx}(x^3)-\dfrac{d}{dx}[\sin(x)] \\\\ &=-4\cdot 3x^2-\cos(x) \\\\ &=-12x^2-\cos(x) \end{aligned}$ In conclusion, $\dfrac{d}{dx}[-4x^3-\sin(x)]=-12x^2-\cos(x)$